266 ANALYTICAL MECHANICS
When /, the moment of inertia, remains constant, as in the case of a rigid body rotating about a fixed axis, the last integration can be performed at once and the following result obtained:
Gd^/w-M). (II)
207. Angular Momentum. —The magnitude Jca is called angular momentum and is defined as the product of the moment of inertia by the angular velocity. Since / is a scalar. 7co is a vector which has the same direction as o>. Equation (II) states that angular impulse equals the change in the angular momentum.
208. Moment of Momentum.--—Angular momentum is often called moment of momentum, because the former may be considered as the moment of the linear momenta of the particles of the system under consideration. Let dm be an element of mass, r its distance from the axis of rotation, and v its linear velocity. Then the moment of the momentum of dm about the axis is r*v dm. Therefore the total moment of momentum is
I r • v dm = I r • rco dm
JQ Jo
dm • co
. r
~Jo
which is the angular momentum.
209. Dimensions and Units. — Substituting the dimensions of G, ^ /, and a in equation (II) we find that both angu .-lar impulse and angular momentum have the dimensio ,ns ]. The units are also the same for both. r The
2
C.G.S. unit is ~—!------ and the British unit is ft. Ib.
sec.